tldr; Use @fortawesome/pro-light-svg-icons, @fortawesome/pro-regular-svg-icons NPM packages.
I ran into this problem today, and I think the main question of how to use Pro version of the icons has not been addressed. The official docs aren't much help. This is what I had to do for using faFilePlus icon, the light version:
import { FontAwesomeIcon } from '@fortawesome/react-fontawesome'
import { faFilePlus } from '@fortawesome/pro-light-svg-icons/faFilePlus';
and then:
<FontAwesomeIcon icon={faFilePlus} />
Note the use of @fortawesome/pro-light-svg-icons NPM package. This is undocumented anywhere in the FA's official docs and I had to dig through several places to arrive at this solution. The official docs only talk about the use of "free" icons but don't say where to find the equivalent NPM package for light, regular etc.
Also note that this does require NPM to be authenticated as mentioned in one of the answers.